Spieth, 20, carded a four-under-par 69, as did the defending champion Dustin Johnson, to be 14 under alongside Simpson, who shot 68.
Zach Johnson could not consolidate the good work he had produced over the previous two days as two bogeys brought him a 74 as he slipped to fourth at 12 under. One stroke further back were a group of four headed by Jason Dufner and Brandt Snedeker, while the Masters champion Adam Scott was the leading non-American at 10 under.
Scotland's Martin Laird slipped a couple of places to 21st despite a 70, while Brian Gay had the round of the day with an eight-under 65. At 12 over, Derek Ernst was the only player in the 30-strong field to be over par.